Monday, October 17, 2005

'Idle Argument' Essay

There are a range of related arguments which fall under the heading of the “Idle Argument”. This essay will discuss those variations which argue from future truths to fatalistic conclusions, i.e. what David Buller calls “the standard argument for fatalism”.[1] The original argument, traceable back to the Stoics,[2] attempted to prove that there is no point seeing a doctor if you are unwell. After all, if you will get well, then there’s no need to see the doctor. And if you won’t get well, then the doctor can’t help you. Either way, action seems unnecessary. We can generalize this reasoning to yield the following argument-schema, for any future event E and a related action Φ:[3]

(P1)If E will occur, then E will occurwhether or not you Φ.

(P2)If E will not occur, then E will not occur whether or not you Φ.

(P3)But either E will occur or it will not.

(C)Therefore, with regard to E, it is futile to Φ.

Intuitively, the problem with such fatalism is that it ignores the causal potency of your actions. Whether E occurs may well depend on whether or not you Φ. If the first two premises are interpreted in such a way as to deny this, then the problem surely lies with them. On this reading, we understand the premises as making modal claims. Suppose you did Φ, and subsequently brought about E. The first premise then claims that E would still have occurred even if you did not Φ. This claim is unmotivated – there is no reason to think that it will generally be true. The second premise will be faulty in the same respect. If we think that Φ-ing would have been sufficient to make E occur, then we will rejectpremise two on the modal reading of “whether or not you Φ”.

But this puzzle is not so easily solved, for there is another interpretation of these premises, according to which they are undeniably true. We obtain this result by interpreting the premises non-modally, as mere material conditionals. Given that E will occur, it follows trivially that E will occur. Indeed, this entailment is valid independently of any other propositions, including the proposition that you Φ. We may take the phrase “whether or not you Φ” to be parenthetical, and interpret the first two premises as instances of the general tautological form: “if p, then p (no matter q)”. Made more logically rigorous, we take this to be equivalent to the logical truth:

p → ((q v ~q) → p)

To fill in the details: ‘q’ is the proposition that you Φ; and we take ‘p’ = ‘E will occur’ for (P1), and ‘E will not occur’ for (P2). So, upon this interpretation, the first two premises are logical truths, and hence undeniable.

This does not force us to accept the absurd conclusion, however, because such weakened premises no longer support the conclusion. That is, the argument becomes invalid. We can see that this is must be so as a matter of form, for you cannot reason from mere tautologies to a substantive conclusion such as that action is futile. To explain this particular case, note that a goal-directed action is ‘futile’ only if the action has no influence on whether or not the goal-state is realized. That is a causal or modal claim, not a merely logical one. The intermediate conclusion that “E will occur, or not,whether or not you Φ” can be obtained only if we understand the phrase “whether or not you Φ” in the parenthetical sense described above, rather than the more intuitive modal sense which we rejected earlier. All that this conclusion really asserts is that E will either occur or not, and that whichever one is the case, other truths must be consistent with this. There are no modal claims being made, and so no basis on which to claim that action is ‘futile’ in the usual sense. In particular, note that the non-modal claim “E will occur (whether or not you Φ)”, as understood here, is entirely consistent with the modal claim that E would not occurunless you Φ-ed.[4]

So the Idle Argument is nothing to worry about. Although it can be interpreted so as to make either its premises true or its logic valid, it cannot achieve both at once. Its unsoundness can be illustrated by the obvious example whereby E will occur, but only because you will in fact Φ. Here it is quite clear that Φ-ing is not a futile action. The difficulty is in tracing this error back into the argument. The most intuitive way to apply the counterexample to the Idle Argument is to say that it shows (P1) to be false. But if the fatalist reinterprets (P1) non-modally, as a logical truth, then the same counterexample instead shows that the Idle Argument is invalid, as its tautological premises are entirely consistent with Φ-ing being a necessary means to achieving E, and thus not futile at all.

The discussion so far, though hopefully helpful in clarifying the underlying problems, has not presented the Idle Argument in its most compelling form. I now want to consider a strengthened version:[5]

(P1’)If we will win the battle, then it is better to attack with a small force.(P2’)If we will lose the battle, then it is better to attack with a small force.

(P3’)Either we will win the battle or we will lose the battle.(C’)So, it is better to attack with a small force.

In this case the conclusion is no longer ‘idleness’, but there remains a clear analogy with the logic of the Idle Argument.[6] This version seems more difficult to refute, however. If we grant that it is more glorious to win with a small force, and that fewer casualties are suffered in losing with a small force, then the first two premises have significant prima facieplausibility. Moreover, as a simple instance of the disjunction-elimination rule, the logic seems clearly valid. But it would be incredible were it possible for us to establish the conclusion (C’) in such an a priori fashion. Surely it is not always better to attack with a small force. Yet here we have an apparently sound argument which claims to prove exactly that.

As before, the flaw in the argument can be highlighted by considering the obvious counterexample: a case whereby it happens to be true that we will win the battle, but only because we will in fact attack with a large force. Since a smaller force would have caused us to lose the battle, the conclusion (C’) is clearly false in this case. Moreover, the premise (P1’) is also false, if understood as a material conditional, for the antecedent is true and yet the consequent false. We will win the battle, as it happens, but it would not be better to attack with a small force, for that would cause us to lose instead.

That’s the simple diagnosis. But, as before, the defender of the argument might appeal to a different interpretation on which the premises come out true. Plausibly, (P1’) should not be understood as a material conditional, but as a more robust connection of some sort. I claimed above that (P1’) is false because attacking with a small force might cause us to lose. The argument’s defender might complain that this objection contradicts the antecedent assumption that we will win the battle. If it is given that we will win, then surely we need not worry about losing. So while it is better to win with a large force than to lose with a small one, this fact doesn’t refute (P1’) as properly understood. It remains true that it is better to win with a small force than with a large one, so if this is all we mean by (P1’), then the premise is true.

To capture this interpretation in a more logically rigorous way, we might say that (P1’) is equivalent to the following claim: of those possible worlds where we win the battle, the best are those where we attack with a small force. Similarly for the second premise: of those possible worlds where we lose the battle, the best are those where we attack with a small force. Given that these two options are exhaustive, as claimed in (P3’), it follows that the best possible worlds are those where we attack with a small force. Let’s call this claim “(B’)”, for ease of reference. The crucial question is now: can we get from (B’) to (C’), or is the battle argument invalid?

Here we are presumably to appeal to the general rule that if the best possible worlds are ones where you Φ, then it is better for you to Φ. But this inference is invalid. The mere fact that the best possible worlds are ones where you Φ is of little help if those worlds are not accessible to you, in the sense of being susceptible to realization. Further, it might also be the case that the worst possible worlds are ones where you Φ. You might find yourself in a situation where those ‘best possible worlds’ are not accessible to you, but the worst ones are. That is, you might face the option of either Φ-ing and ending up in a terrible situation, or not Φ-ing and remaining in a mediocre situation. Clearly, in such a case it is not better for you to Φ.

For a more concrete example, suppose that, so far as my finances are concerned, the best possible world is one where I win the lottery. That is, a world where I buy a lottery ticket. But in most worlds where I buy a lottery ticket, I lose, and so have wasted my money. So it would be a mistake to say categorically that it is ‘better’ for me to buy a ticket. The mere fact that I do so in the ‘best’ world is insufficient to reach that conclusion. In assessing a course of action, we must consider not only the possible benefits, but also the possible costs. So there is no straightforward inference from (B’) to (C’), and hence the battle argument is invalid. When we interpret the premises as other than material conditionals, though they might then be true, they fail to establish the conclusion.

This might seem puzzling, as the argument appears to be a straightforward instance of the valid argument form:

(P → R)

(Q → R)

(P v Q)

_______

R

But this argument form involves material conditionals, which would render the first two premises of the battle argument false, as was earlier established. On the new interpretation, it is not clear in what sense those premises are really conditionals at all. Instead, they make comparative claims, to the effect that winning-with-a-small-force is better than winning-with-a-large-force, and that losing-with-a-small-force is better than losing-with-a-large-force. The argument then appears to take the form of a proof by cases, showing that in each possible case it is better to attack with a small force, and thus establishing the conclusion (C’). However, the argument fails to actually consider all possible cases. In particular, it fails to compare winning-with-a-large-force to losing-with-a-small-force. This is a serious oversight, given that these could very well be the options open to us. Since this possibility has not been accounted for, we cannot categorically conclude that it would be better to attack with a small force.

One might try to restore the valid form of the argument, whilst retaining true premises, by reinterpreting (P1’) to mean something like, “if it is guaranteed that we will win the battle, then it is better to attack with a small force”, and similarly for (P2’). Again, depending on how we interpret the ‘guarantee’ here, the argument might be made either logically valid or containing true premises, but not both at once. The conditional premises will be true if the ‘guarantee’ has modal implications, i.e. that we would win the battle even with a small force, or – in case of (P2’) – lose it even with a larger one. But it is not true that we are either guaranteed to win or else guaranteed to lose, in this sense, so the argument fails to consider all cases. It fails to take into account those cases where we will win only if we take a large force, for example. So the argument is invalid.[7] Alternatively, if the ‘guarantee’ is non-modal, merely requiring that it be metaphysically fixed that we will win in the actual world, then (P1’) is simply false. It might be “guaranteed” that we will win only because it is also “guaranteed” that we will attack with a large force; and in those closest possible worlds where we do otherwise, we might well lose. If that were so, then it would be false to claim that it is better to attack with a small force. So, either way, the argument fails to establish its conclusion.

The key to understanding both the Idle Argument and its “Battle” variation, is the ambiguity found in the first two premises. They might be interpreted so as to come out true, or else they might be interpreted so as to instantiate a valid argument form. If we equivocate between the two interpretations, then we would seem to have a valid argument with true premises, which would establish the truth of the absurd conclusion. But the puzzle can be dispelled by identifying this equivocation. The premises are false if interpreted one way, and the inference invalid otherwise. Either way, the argument is unsound.

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